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a2ee19a | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 | import OAI.Probability.DilutedSpin.FunctionalContinuity
import OAI.Probability.SATComputability.RationalExponents
namespace OAI
namespace FixedClauseThreshold.Computability
open DilutedSpinGlass _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Filter Set
open scoped Topology BigOperators NNReal
local instance exponentContinuityMeasurable (X : TopCat) : MeasurableSpace X := borel X
local instance exponentContinuityBorel (X : TopCat) : BorelSpace X := ⟨rfl⟩
abbrev UnitExponents (r : ℕ) := {m : Fin r → ℝ // ∀ i, 0 < m i ∧ m i < 1}
def exponentTail {r : ℕ} (m : UnitExponents (r+1)) : UnitExponents r :=
⟨fun i => m.val i.succ, fun i => m.property i.succ⟩
theorem exponentTail_continuous (r : ℕ) :
Continuous (exponentTail (r := r)) := by
apply Continuous.subtype_mk
exact continuous_pi (fun i => (continuous_apply i.succ).comp continuous_subtype_val)
theorem logMean_continuous_exponents {ι : Type} [Fintype ι]
(r : ℕ) (g : (ι → ℝ) → ℝ) (hg : Continuous g)
{B : ℝ} (hB : ∀ x, |g x| ≤ B) (eta : ι → Hierarchy r) :
Continuous (fun m : UnitExponents r => logMean r g m.val eta) := by
induction r with
| zero => exact continuous_const
| succ r ih =>
let (i : ι) : IsProbabilityMeasure ((eta i).toMeasure) := by
exact (eta i : ProbabilityMeasure (Hierarchy r)).property
have hBn : 0 ≤ B := (abs_nonneg (g (fun _ => 0))).trans (hB _)
have hc : Continuous (fun m : UnitExponents (r+1) =>
∫ x : ι → Hierarchy r, Real.exp (m.val 0 * logMean r g (fun i => m.val i.succ) x)
∂Measure.pi (fun i => (eta i).toMeasure)) := by
apply continuous_of_dominated (bound := fun _ => Real.exp B)
· intro m
exact (Real.continuous_exp.comp ((logMean_continuous_bound r g hg hB _
(fun i => (m.property i.succ).1)).1.const_mul (m.val 0))).aestronglyMeasurable
· intro m
apply ae_of_all
intro x
rw [Real.norm_eq_abs, abs_of_pos (Real.exp_pos _)]
apply Real.exp_le_exp.mpr
have hb := (logMean_continuous_bound r g hg hB _
(fun i => (m.property i.succ).1)).2 x
calc
_ ≤ m.val 0 * B := mul_le_mul_of_nonneg_left (le_abs_self _ |>.trans hb) (m.property 0).1.le
_ ≤ B := by nlinarith [(m.property 0).2]
· exact integrable_const _
· apply ae_of_all
intro x
exact Real.continuous_exp.comp (((continuous_apply 0).comp continuous_subtype_val).mul
((ih x).comp (exponentTail_continuous r)))
have hpos (m : UnitExponents (r+1)) :
0 < ∫ x : ι → Hierarchy r,
Real.exp (m.val 0 * logMean r g (fun i => m.val i.succ) x)
∂Measure.pi (fun i => (eta i).toMeasure) :=
MeasureMean.exp_integral_pos _
(logMean_continuous_bound r g hg hB _ (fun i => (m.property i.succ).1)).1.measurable
(logMean_continuous_bound r g hg hB _ (fun i => (m.property i.succ).1)).2 _
exact (hc.log (fun m => (hpos m).ne')).div
((continuous_apply 0).comp continuous_subtype_val) (fun m => (m.property 0).1.ne')
theorem trialLog_continuous_exponents {ι : Type} [Fintype ι]
(r : ℕ) (g : (ι → ℝ) → ℝ) (hg : Continuous g)
{B : ℝ} (hB : ∀ x, |g x| ≤ B) (ζ : Hierarchy (r+1)) :
Continuous (fun m : UnitExponents r => trialLog r ζ m.val g) := by
let : IsProbabilityMeasure ζ.toMeasure := (ζ : ProbabilityMeasure (Hierarchy r)).property
apply continuous_of_dominated (bound := fun _ => B)
· intro m
exact (logMean_continuous_bound r g hg hB m.val (fun i => (m.property i).1)).1.aestronglyMeasurable
· intro m
exact ae_of_all _ (fun eta => by
simpa only [Real.norm_eq_abs] using
(logMean_continuous_bound r g hg hB m.val (fun i => (m.property i).1)).2 eta)
· exact integrable_const _
· exact ae_of_all _ (logMean_continuous_exponents r g hg hB)
theorem functional_continuous_unit_exponents {p r : ℕ} (M : Model p)
(hθ : Integrable (fun z : InteractionSample p => ‖z.1‖) M.disorder.toMeasure)
(hh : Integrable (fun h : ℝ => |h|) M.field.toMeasure)
(ζ : Hierarchy (r+1)) :
Continuous (fun m : UnitExponents r => functional M r ζ m.val) := by
have hcfield (k : ℕ) (θ : Fin k → InteractionSample p) :
Continuous (fun m : UnitExponents r => trialSiteField M m.val ζ θ) := by
apply continuous_of_dominated (bound := fun h : ℝ => |h|+∑ j, ‖(θ j).1‖)
· intro m
exact (integrable_trial_site_field M m.val (fun i => (m.property i).1) hh ζ θ).aestronglyMeasurable
· intro m
exact ae_of_all _ (fun h => by
simpa only [Real.norm_eq_abs] using (trial_site_bound r m.val (fun i => (m.property i).1) ζ θ h))
· exact hh.add (integrable_const _)
· exact ae_of_all _ (fun h => trialLog_continuous_exponents r _
(continuous_siteLog θ h) (siteLog_bound θ h) ζ)
have hcdisorder (k : ℕ) :
Continuous (fun m : UnitExponents r => trialSiteDisorder M m.val ζ k) := by
apply continuous_of_dominated (bound := fun θ : Fin k → InteractionSample p =>
(∫ h : ℝ, |h| ∂M.field.toMeasure)+∑ j, ‖(θ j).1‖)
· intro m
exact (stronglyMeasurable_trialSiteField M m.val (fun i => (m.property i).1) ζ k).aestronglyMeasurable
· intro m
exact ae_of_all _ (fun θ => by
simpa only [Real.norm_eq_abs] using (trialSiteField_bound M m.val (fun i => (m.property i).1) hh ζ θ))
· exact (integrable_const _).add (interaction_sum_integrable M hθ k)
· exact ae_of_all _ (hcfield k)
have hcsite : Continuous (fun m : UnitExponents r =>
∫ k, trialSiteDisorder M m.val ζ k ∂poissonMeasure (M.alpha*p)) := by
apply continuous_of_dominated (bound := fun k : ℕ =>
(∫ h : ℝ, |h| ∂M.field.toMeasure)+k*(∫ z : InteractionSample p, ‖z.1‖ ∂M.disorder.toMeasure))
· intro m
exact (measurable_of_countable _).aestronglyMeasurable
· intro m
exact ae_of_all _ (fun k => by
simpa only [Real.norm_eq_abs] using (trialSiteDisorder_bound M m.val (fun i => (m.property i).1) hθ hh ζ k))
· exact (integrable_const _).add ((poisson_integrable_count _).mul_const _)
· exact ae_of_all _ hcdisorder
have hcedge : Continuous (fun m : UnitExponents r =>
∫ θ : InteractionSample p, trialLog r ζ m.val (fun x => Real.log (edge θ.1 x))
∂M.disorder.toMeasure) := by
apply continuous_of_dominated (bound := fun θ : InteractionSample p => ‖θ.1‖)
· intro m
exact (((trial_edge_lipschitz r m.val (fun i => (m.property i).1) ζ).continuous.comp
continuous_fst).aestronglyMeasurable)
· intro m
exact ae_of_all _ (fun θ => by
simpa only [Real.norm_eq_abs] using (trial_edge_bound r m.val (fun i => (m.property i).1) ζ θ.1))
· exact hθ
· exact ae_of_all _ (fun θ => trialLog_continuous_exponents r _
(continuous_log_edge θ.1) (log_edge_bound θ.1) ζ)
exact (continuous_const.add hcsite).sub (continuous_const.mul hcedge)
theorem isOpen_unit_exponents (r : ℕ) :
IsOpen {m : Fin r → ℝ | ∀ i, 0 < m i ∧ m i < 1} := by
have he : {m : Fin r → ℝ | ∀ i, 0 < m i ∧ m i < 1} =
⋂ i : Fin r, {m | 0 < m i ∧ m i < 1} := by ext; simp
rw [he]
exact isOpen_iInter_of_finite (fun i =>
(isOpen_lt continuous_const (continuous_apply i)).inter
(isOpen_lt (continuous_apply i) continuous_const))
theorem functional_continuousAt_exponents {p r : ℕ} (M : Model p)
(hθ : Integrable (fun z : InteractionSample p => ‖z.1‖) M.disorder.toMeasure)
(hh : Integrable (fun h : ℝ => |h|) M.field.toMeasure)
(ζ : Hierarchy (r+1)) {m : Fin r → ℝ} (hm : Exponents m) :
ContinuousAt (fun m => functional M r ζ m) m := by
have hc : ContinuousOn (fun m => functional M r ζ m)
{m : Fin r → ℝ | ∀ i, 0 < m i ∧ m i < 1} :=
continuousOn_iff_continuous_domRestrict.mpr
(functional_continuous_unit_exponents M hθ hh ζ)
exact (hc m hm.2).continuousAt ((isOpen_unit_exponents r).mem_nhds hm.2)
end FixedClauseThreshold.Computability
end OAI
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